Definition:Möbius Strip

Definition

A Möbius strip is a surface with boundary obtained by twisting a side of a rectangle by $180$ degrees, then joining the twisted side and the opposite side of it together.

Thus in the above diagram, $AB$ is identified with $CD$.

Formal Construction

Let $T$ be the square embedded in the Cartesian plane defined as:

$T = \closedint 0 1 \times \closedint 0 1$

Let $T'$ be the quotient space formed from $T$ using the identification mapping $p: T \to T'$ as follows:

$\forall \tuple {x, y} \in T: \map p {x, y} = \begin {cases} \tuple {1, 1 - y} & : x = 0 \\ \paren {x, y} & : x \ne 0 \end {cases}$

Also known as

Some sources (particularly American) use the term Möbius band on the grounds that the word strip may be considered inappropriate language to use in front of children and adolescents.

Also see

• Möbius Strip has Euler Characteristic Zero

• Results about the Möbius strip can be found here.

Source of Name

This entry was named for August Ferdinand Möbius.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.8$: Quotient spaces
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Möbius strip
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Möbius strip (Möbius band)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Möbius strip (Möbius band)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Möbius band (strip)

• Weisstein, Eric W. "Möbius Strip." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MoebiusStrip.html